PROOFS. Which one, when.

Direct Proof.
Proving the Contrapositive.
Proof by Counterexample.
Proof by Contradiction.

How do you differentiate when to use each of these?

Shouldn't you use the most obvious to you for that particular problem?

Depends really.

I tend to use counterexamples a lot more for analysis type things, because there's often some nice pathologic examples that work for a lot of things, for example if you're talking about Banach spaces then l_inf or l_0 are often great for counterexamples because they're not reflexive, or because l_inf isn't seperable etc. If you do enough questions you tend to see which ones are good to use, spike functions etc.

If you know the proof's going to be just basic manipulation and applying definitions, but it might get complicated, then it's generally a toss up between direct proof and contrapositive. One's going to be easier, but it's not normall extremely obvious to tell.

All I can say is, do lots of proofs, you'll get a feel for it.

There's no magical handbook that tells you how to prove a given theorem. But here are some fairly common rules of thumb:

First, note that "proving the contrapositive" and "proof by contradiction" are identical. You use this method generally when you're trying to prove something of the form X -> Y where ~Y gives you some useful information to work with. For example, "if k is a prime number greater than 2, k is odd." Assume ~"k is odd" (ie, k is even), then k must be divisible by 2, and since k > 2, k is necessarily not prime.

Direct proof is commonly used when X gives you some useful information to work with. For example, "if f(x) is a differentiable even function, then f'(0) = 0." Using the fact that f is an even function, we can say lim h->0 (f(h) - f(-h))/h = lim h->0 (f(h) - f(h))/h = 0, so f'(0) = 0.

Proof by counterexample is useful when you're trying to prove something like ~X where X contains a "for all" quantifier. This is generally used in lower level university math classes when disproving something, for example "if k is prime, k is odd." Clearly k = 2 is a counterexample to that.

And, as >>3 said, do lots of proofs and you'll get a feel for it.

Proof by construction is ALWAYS the best, if possible. (If you're trying to prove something exists, that is)

Direct deduction from previously known theorems is next best.

Contradiction is usually easiest, but is graceless and should be avoided if reasonably possible.  Contradiction shows why something can't be false, but it doesn't show why it must be true.

Counterexamples should be used in disputing a claim. When I write a proof for a given statement, I'll first analyze it, checking special cases for counterexamples. Generally, by the time I finish with special cases, either I've found a counterexample, or I'll have a good idea of how and why the concept works. Generally I'll just try and prove it directly. I only rarely have to use the contrapositive, but that's an option if I feel like I'm getting stonewalled on a direct proof. Typically, if I can't seem to figure out the critical step for a direct proof, I'll do proof by contradiction, a sort of 'last resort' proof. Don't ask me what it is, but most mathematicians generally seem to see proof by contradiction as not being as elegant as a direct proof. Something about it feels superficial.